Chapter 20: Structured Universe (Cosmic Web)

20.2 Acoustic Oscillations

The spots in the CMB are not random; they form a highly structured, rhythmic pattern of peaks and troughs in the Angular Power Spectrum. This section derives the physics of the primordial "sound waves" that vibrated through the early universe, detailing how the competition between emergent forces sculpted the acoustic peaks.

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20.2.1 Lemma: Gravitational and Entropic Competing Forces

Interplay of Attractive Ollivier-Ricci Compression and Radiative Restoring Forces in Primordial Plasma

• Attractive Compression: Primordial overdensities ($\delta\rho_3 > 0$) generate an attractive force $F_g \propto -\nabla \rho_3$ (emergent gravity), compressing the baryon-photon plasma inward.
• Entropic Restoring Force: As the plasma compresses, the local density of photon motifs spikes. To maximize entropy, the rewrite rules favor scattering updates that disperse the photons outward, generating a powerful pressure force: $F_p = -\nabla P$.
• Standing Sound Waves: The competition between gravitational compression and radiative entropic expansion creates standing sound waves in the plasma. The peaks correspond to modes captured at maximum compression or rarefaction at the moment of last scattering.

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20.2.2 Postulate: Sterile Braid Scaffolding

Anchoring of Gravitational Potential Wells by Electromagnetically Inert Sterile Braid Structures

• Dark Matter Relics: The dark sector consists of "sterile braids" (§21.1)—braid topologies that possess rest mass complexity ($C[\beta]$) but lack the electroweak twists/rungs required to couple to electromagnetic photon motifs.
• Shadow Scaffolding: Because they lack charge topology, these sterile braids do not interact with photons and remain unaffected by radiation pressure.
• Oscillation Anchors: While the baryonic plasma oscillates violently, the sterile braids remain stationary, forming a stable gravitational potential scaffolding that guides and amplifies the baryonic sound waves.

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20.2.3 Theorem: Angular Power Spectrum Peaks

Spacing of Acoustic Peak Coordinates in Angular Power Spectrum via Sound Horizon Scale

• Sound Horizon Boundary: The sound waves can only travel a finite distance before recombination, defining the Sound Horizon scale: $r_s(t_*) = \int_0^{t_*} c_s(t) dt$
• Angular Power Peaks: The acoustic peak positions in the angular power spectrum correspond to multiples of the sound horizon projected onto the sky.
• Braid Composition Signature: The relative heights of the peaks are uniquely determined by the ratio of baryonic braids to sterile dark matter braids.

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20.2.4 Proof: Angular Power Spectrum Peaks

Verification of Acoustic Peaks through Direct Numerical Solution of Sound Horizon Integrals

• Horizon Scale Evaluation: The proof calculates the sound horizon scale using the emergent speed of sound $c_s = 1/\sqrt{3(1 + 3\rho_b/4\rho_\gamma)}$.
• Spectrum Verification: It mathematically derives the peak locations $l_m \approx m \pi D_A / r_s$, verifying that they match the observational coordinates measured by the Planck satellite, confirming the existence of the non-baryonic sterile braid species.